1. Field of the Invention
The present invention generally relates to a method and an apparatus for optical frequency measurement and, more particularly, to a method and an apparatus using one or two frequency-stabilized mode-locked laser combs operating at different repetition rates to measure the frequency of a laser under measurement (LUM). The ordinal comb number difference can be obtained according to the beat frequencies, the offset frequencies and the repetition rates so as to measure the ordinal comb number and the frequency of the LUM.
2. Description of the Prior Art
Since 1999 T. W. Hänsch used femtosecond pulsed lasers in optical frequency measurement of the cesium D1 line, mode-locked lasers have attracted lots of attention in optical frequency measurement.
As shown in FIG. 1, a mode-locked laser is an optical frequency comb (OFC) composed of a plurality of comb lines with identical frequency pitches in frequency-domain. The frequency of each comb line is integer multiples of the repetition rate fr plus an offset frequency fo. That is, the frequency fn of the nth comb line can be expressed as:fn=n×fr+fo 
wherein n is a positive integer, fr is the pulse repetition frequency (or, in brief, the repetition rate) and fo is the carrier-envelope offset frequency (or, in brief, the offset frequency). The offset frequency fo is usually measured by using the self-referencing technique. FIG. 2 shows the theory of the f−2f self-referencing technique.
However, two possible values fo and fr−fo may be obtained between 0 and fr with the self-referencing technique to measure the offset frequency of a mode-locked laser. It is required to determine one of fo and fr−fo to be the correct offset frequency. By definition, either fo or fr−fo plus the integer multiples of the repetition rate fr can be defined as an offset frequency. Moreover, for a measured beat frequency fb, the beat frequency fb occurs when the frequency fL of the LUM is fb higher or lower than the comb line. Therefore, the frequency of the LUM can be fL=n×fr+fo±fb or fL=n×fr+(fr−fo)±fb=(n+1)×fr−fo±fb. Since n is left undetermined, the frequency of the LUM fL can be expressed as:fL=n×fr±fo±fb 
In the latter representation, the offset frequency can be −fo. Even though the ordinal comb numbers obtained using the two aforementioned methods may be different, the frequencies of the LUM thus measured are identical. In the specification of the present invention, the frequency of each comb line is integer multiples of the repetition rate fr plus a positive offset frequency fo, which is the same as the former representation.
The correct offset frequency and the sign of the beat frequency can be determined by changing the repetition rate or the offset frequency according to the corresponding variation of the beat frequency. The ordinal comb number can be determined based on the following methods. In the first method, historic measurement data of the frequencies of the LUM are used, wherein the n value is determined only when the precision is within ±fr/4. In the second method, a wavemeter is used to determine an approximate frequency of the LUM. If the precision of the frequency measured by the wavemeter is within ±fr/4, it can be decided which comb line is closest to the frequency of the LUM so as to determine the n value. The precision of a commercial wavemeter is about 2×10−7, which leads to 40 MHz inaccuracy for a 1550 nm laser. Therefore, the pitch of the OFC is at least 160 MHz so that it can work with the commercial wavemeter for optical frequency measurement.
In the third method, Long-Sheng Ma et al. used an OFC to determine the ordinal comb number without using a wavemeter by measuring the beat frequencies at multiple repetition rates fr and fr′ and recording the ordinal comb number variation m when the repetition rate changed. Accordingly, they derived an equation for calculating the ordinal comb number, which is expressed as:n=[±fo′−(±fo)+m fr′±fb′−(±fb)]/(fr−fr′)  (1)
wherein m is the ordinal comb number variation when the repetition rate changes from fr to fr′; fb, fb′ are the beat frequencies for the LUM and the OFC; fr, fr′ and fo, fo′ are the repetition rates and offset frequencies before and after the OFC is adjusted, respectively. At that time, the signs ± could not be determined, and therefore, the correct ordinal comb number could be obtained only after at least two m values were compared. Moreover, in this method, the ordinal comb number variation was recorded as the repetition rate changed. For low repetition rate laser, the number of recorded ordinal comb number is very large. Therefore, this method is not practical.
In the three foregoing methods, sufficient historic data, a high-precision wavemeter and gradually changing repetition rates are required to record all of the ordinal comb number variations so as to obtain the frequency of the LUM. Obviously, these methods are not explicit and not suitable for use in optical frequency measurement for all repetition rates.
Jin-Long Peng et al. disclosed a method for measuring the ordinal comb number by using mode-locked lasers, which is useful for a wide range of repetition rate. First, the sign of the beat frequency was determined according to the change of the repetition rate and the change of the beat frequency. Then the offset frequency was changed and the correct offset frequency of the mode-locked laser was determined according to the change of the beat frequency, as shown in FIG. 3A and FIG. 3B. The ordinal comb number was determined using two mode-locked laser combs operated at different repetition rates to generate the beat frequencies with the LUM. The frequency of the LUM can be expressed as:fL=n fr1+fo1±fb1  (2)fL=(n+m)fr2+fo2±fb2  (3)
From Equations (2) and (3), n can be derived asn=[m fr2+fo2−fo1±fb2−(±fb1)]/(fr1−fr2)  (4)
In Equation (4), fr1, fr2, fo1, fo2, fb1, fb2 can be measured by a microwave frequency counter. The correct offset frequency fo1 and fo2 and the signs of fb1 and fb2 can be determined using the foregoing methods. Equation (4) is similar to the Equation (1) derived by Long-Sheng Ma et al. (L.-S. Ma, M. Zucco, S. Picard, L. Robertsson, and R. S. Windeler, IEEE Journal of Selected Topics in Quantum. Electronics. 9, 1066 (2003)) even though the offset frequency is expressed by a different notation. Moreover, Jin-Long Peng et al. disclosed a method for measuring the ordinal comb
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                                             f                                                  r                          ⁢                                                                                                          ⁢                          2                                                                                                                                    f                                                  r                          ⁢                                                                                                          ⁢                          1                                                                    /                      n                                                        +                                                                                    n                        ⁡                                                  (                                                                                    f                                                              r                                ⁢                                                                                                                                  ⁢                                1                                                                                      -                                                          f                                                              r                                ⁢                                                                                                                                  ⁢                                2                                                                                                              )                                                                    2                                                                                      f                                                  r                          ⁢                                                                                                          ⁢                          1                                                                    ⁢                                              f                                                  r                          ⁢                                                                                                          ⁢                          2                                                                                                      -                                                                                    f                                                  o                          ⁢                                                                                                          ⁢                          2                                                                    -                                                                        f                                                      o                            ⁢                                                                                                                  ⁢                            1                                                                          ±                                                                              f                                                          b                              ⁢                                                                                                                          ⁢                              2                                                                                ⁢                          μ                          ⁢                                                                                                          ⁢                                                      f                                                          b                              ⁢                                                                                                                          ⁢                              1                                                                                                                                                                  f                                              r                        ⁢                                                                                                  ⁢                        2                                                                                                                                                                    ≈                                ⁢                                                                                                    f                                                  r                          ⁢                                                                                                          ⁢                          1                                                                    -                                              f                                                  r                          ⁢                                                                                                          ⁢                          2                                                                                                                                    f                                                  r                          ⁢                                                                                                          ⁢                          1                                                                    /                      n                                                        -                                                                                    f                                                  o                          ⁢                                                                                                          ⁢                          2                                                                    -                                                                        f                                                      o                            ⁢                                                                                                                  ⁢                            1                                                                          ±                                                                              f                                                          b                              ⁢                                                                                                                          ⁢                              2                                                                                ⁢                          μ                          ⁢                                                                                                          ⁢                                                      f                                                          b                              ⁢                                                                                                                          ⁢                              1                                                                                                                                                                  f                                              r                        ⁢                                                                                                  ⁢                        2                                                                                                                                ⁢                                  ⁢                  if          ⁢                                          ⁢                                                    n                ⁡                                  (                                                            f                                              r                        ⁢                                                                                                  ⁢                        1                                                              -                                          f                                              r                        ⁢                                                                                                  ⁢                        2                                                                              )                                            2                                                      f                                  r                  ⁢                                                                          ⁢                  1                                            ⁢                              f                                  r                  ⁢                                                                          ⁢                  2                                                              ⁢                      <<            1                                              (        5        )            number difference m, which was derived from Equation (4) and expressed as:
The correct offset frequency and the signs of the beat frequencies could be determined using the foregoing method, wherein the repetition rate variation required for changing an ordinal comb number was used to measure fr1/n, as shown in FIG. 4A to FIG. 4D, to calculate the m value using Equation (5). Such a method is easier than the method disclosed by Long-Sheng Ma et al. because it is not required to gradually record the change of the ordinal comb number. However, the repetition rate needs to be gradually changed to determine whether the change of the ordinal comb number is completed according to whether the beat frequency returns to its original value twice. This method is time-consuming and an automatized apparatus for optical frequency measurement based on this method is not easy to construct for rapid optical frequency measurement.
Therefore, there is need in providing a method and an apparatus for measuring the ordinal comb number variation without scanning the repetition rate so as to obtain the ordinal comb number and the frequency of an LUM.